IJMMS 2003:72, 4517–4538
PII. S0161171203302108
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN
INEQUALITIES ASSOCIATED TO A MINIMAX
PROBLEM WITH ADDITIVE FINAL COST
SILVIA C. DI MARCO and ROBERTO L. V. GONZÁLEZ
Received 11 February 2003
We study a minimax optimal control problem with finite horizon and additive final
cost. After introducing an auxiliary problem, we analyze the dynamical programming principle (DPP) and we present a Hamilton-Jacobi-Bellman (HJB) system. We
prove the existence and uniqueness of a viscosity solution for this system. This
solution is the cost function of the auxiliary problem and it is possible to get the
solution of the original problem in terms of this solution.
2000 Mathematics Subject Classification: 49L20, 49L25, 49K35, 49K15.
1. Introduction. The optimization of dynamic systems where the criterion
is the maximum value of a function is a frequent problem in technology, economics, and industry. This problem appears, for example, when we attempt to
minimize the maximum deviation of controlled trajectories with respect to a
given “model” trajectory. Minimax problems differ from those usually considered in the optimal control literature where a cumulative cost is minimized.
Since in some cases, minimax problems describe more appropriately decision
problems arisen in controlled systems whose performance is evaluated with a
unique scalar parameter, the minimax optimization has received considerable
attention in recent publications (see, e.g., [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16,
17, 18, 19, 22]). Furthermore, minimax problems are related to the design of
robust controllers (see [14]).
In addition, from the academic point of view, the minimax optimal control problem is of interest in the area of game theory because minimax problems can be seen as a game (see [18]) where a player applies ordinary controls and the other one—using complete and privileged information—chooses
a stopping time. Problems of this type lead to the treatment of nonlinear
partial differential inequalities akin to those appearing in the obstacle problem (with obstacle given in explicit or implicit form, see [10]). To find solutions of these systems, we must consider generalized solutions—even discontinuous solutions—since commonly there do not exist classical solutions
of such systems (see [2, 3]). The treatment of the infinite horizon problem
also presents great analytical difficulties because the optimal cost is neither
necessarily lower semicontinuous nor upper semicontinuous. Moreover, the
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S. C. DI MARCO AND R. L. V. GONZÁLEZ
optimal cost cannot be approximated with a sequence of finite horizon problems. Studies concerning these issues can be seen in [17, 19]. Besides, it is
also important to develop numerical methods to compute these solutions in
an approximate way because in general it is not possible to find exact analytical solutions. Numerical methods to obtain approximated open-loop optimal
controls are analyzed in [24] and for closed loop, see [8, 15].
Here, we analyze a minimax optimal control problem where the functional
to be optimized not only depends on the maximum of a function along the
complete trajectory of the system but also it takes into account (in an additive
fashion) another function of the final state of the system.
For a problem with such a structure, a dynamical programming principle
cannot be formulated merely in terms of the initial time and the initial state.
To obtain a DPP, we introduce an auxiliary parameter which “remembers” the
past maximum values (the use of a similar procedure can be seen in [4, 5]).
Using this parameter, we present an auxiliary problem which gives the solution of the original problem when a particular value of the auxiliary parameter
is chosen. For this second optimal control problem, we establish the associated dynamical programing principle (DPP) and a Hamilton-Jacobi-Bellman
(HJB) equation. Finally, we prove that the optimal cost of the auxiliary problem
is the unique solution of the associated HJB equation.
The paper is organized as follows. In Section 2, we present the optimization
problem. In Section 3, we describe the auxiliary problem and its relation with
the original one. We also establish there the dynamical programming equation.
In Section 4, we give the HJB equation associated to the problem and we prove
the uniqueness of the solution in the viscosity sense.
2. The optimization problem
2.1. Presentation of the problem. We consider a minimax optimal control
problem with finite horizon and final cost. More specifically, the problem consists in minimizing the functional J,
t, x, α(·) → J t, x, α(·) ,
J : [0, T ] × Rm × Ꮽ → R,
J t, x, α(·) = Ψ yα (T ; x, t) + ess sup f s, yα (s; x, t), α(s) ,
(2.1)
(2.2)
s∈[t,T )
where yα (·; t, x) represents the state of a dynamic system which evolves from
the pair (t, x) according to the following differential equation:
yα (s; t, x) = g s, yα (s; t, x), α(s) a.e. s ∈ [t, T ],
(2.3)
yα (t; t, x) = x, x ∈ Rm .
To simplify the notation, and whenever this simplification does not produce
misunderstandings, we will write directly yα (·; t, x) = y(·). The set Ꮽ = L∞ ((0,
T ); A) is the set of admissible control policies and A is the control set.
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
4519
The optimal cost function is given by
u(t, x) = inf J t, x, α(·) .
α(·)∈Ꮽ
(2.4)
This problem is an extension of another one analyzed by Barron and Ishii [10]
and by Di Marco and González [15, 18]. The problem with an additive final
cost considered here can represent scenarios where the performance of the
applied control is measured jointly (in an additive fashion) by the maximum
of a function along the trajectory and by a function of the final state.
To give an example of this type of problems, we consider the following economic case.
Elements of the problem.
(1) The vector y(t) of economic activities (manufacturing, services, etc.);
(2) GDP(y(T )), gross domestic product at the end of the period [0, T ];
(3) a(t), policy of resources assignments;
(4) f (y(t), a(t)), unemployment level.
We consider the following functional, which measures the effectiveness of
the economic policy. The functional measures both the positive aspects (the
GDP) and the negative ones (the unemployment level f (y(t), a(t))) of the economic policy in the following form:
J a(·) = GDP y(T ) / max f y(t), a(t) .
t∈[0,T ]
(2.5)
The maximization of J is equivalent to the maximization of the functional
log GDP y(T ) − max log f y(t), a(t) .
t∈[0,T ]
(2.6)
Hence, if we define
Ψ y(T ) = − log GDP y(T ) ,
(2.7)
the problem is equivalent to the minimization of the functional
Ψ y(T ) + max log f y(t), a(t) ,
t∈[0,T ]
(2.8)
and so we must deal with a problem of type (2.2).
As we will see, the new problem presents an additional difficulty because it
is not possible to establish a DPP only in terms of the variables (t, x). To avoid
this difficulty, we introduce an auxiliary problem which generalizes the problem presented above and we establish the DPP corresponding to the auxiliary
problem. We also present an HJB equation associated to the optimal cost. Finally, using a methodology similar to that one presented in [20, 23], we prove
that the optimal cost is the unique solution of this HJB equation.
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S. C. DI MARCO AND R. L. V. GONZÁLEZ
2.2. General assumptions. Let BUC([0, T ] × Rm × A) be the set of bounded
and uniformly continuous functions in [0, T ] × Rm × A. We assume that the
following hypotheses hold:
(H1 ) g : [0, T ] × Rm × A Rm , g ∈ BUC([0, T ] × Rm × A), and
g(t, x, a) ≤ Mg ,
g(t, x, a) − g t̂, x̂, a ≤ Lg t − t̂ + x − x̂ ,
(2.9)
for all t, t̂ ∈ [0, T ], all x, x̂ ∈ Rm , and all a ∈ A;
(H2 ) f : [0, T ] × Rm × A R, f ∈ BUC([0, T ] × Rm × A), and
mf ≤ f (t, x, a) ≤ Mf ,
f (t, x, a) − f t̂, x̂, a ≤ Lf t − t̂ + x − x̂ ,
(2.10)
for all t, t̂ ∈ [0, T ], all x, x̂ ∈ Rm , and all a ∈ A;
(H3 ) Ψ : Rm R, Ψ ∈ BUC(Rm ), and
Ψ (x) − Ψ x̂ ≤ LΨ x − x̂ ,
∀x, x̂ ∈ Rm ;
(2.11)
(H4 ) A is compact in Rq .
Note 2.1. The above-stated hypotheses are not the minimal ones under
which the principal results of this paper hold. In particular, the Lipschitz continuity of f and Ψ can be replaced by the uniform continuity. We have used
(H1 ), (H2 ), (H3 ), and (H4 ) in order to simplify the proof of the central results.
3. Auxiliary problem. In the problem presented above, it is not possible
to establish a DPP merely in terms of the variables (t, x) because the optimal
control policy generally depends not only on the current state (t, x) but also
on the past values of the trajectory. We see the following example.
Example 3.1. We consider (t, x) ∈ [0, 1] × [−2, 2]. The control set is A =
[−0.5, 0.5]. The instantaneous cost f , the final cost Φ, and the dynamic g are
given by
Φ(x) = x,
f (t, x, a) = (t − 0.5)− + a− ,
g(t, x, a) = a 1.5 − 0.5|x| − 0.5 |x| − 1 .
(3.1)
Figure 3.1 shows the optimal trajectories corresponding to a couple of different initial points (t, x).
So, the optimal control policy depends not only on the current state (t, x)
but also on the past values of the trajectory.
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
4521
0.5
The position evolution (y)
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
The time of evolution (t)
Figure 3.1
3.1. Auxiliary variable and problem reformulation. In order to develop
the analytical and numerical treatment of the problem, we extend the state of
the system introducing an auxiliary state ym+1 (·) (which is actually ym+1 (·) :=
yα,m+1 (·, t, x, ρ)). To simplify the notation, we will define
hα (t, τ) := ess sup f s, y(s), α(s) ,
τ ∈ (t, T ].
(3.2)
s∈[t,τ)
The auxiliary variable takes the following form, for all τ ∈ [t, T ],

ρ,
yα,m+1 (τ, t, x, ρ) =
max ρ, h (t, τ),
α
if τ = t,
if τ ∈ (t, T ].
(3.3)
Note 3.2. The additional variable ym+1 “stores” the maximum value of the
function f from the initial time t to the current time τ when its initial value
is suitably chosen. More clearly, by considering ρ ≤ mf , from (3.3), it follows
that
ym+1 (T ) = hα (t, T ).
(3.4)
Note 3.3. When ρ ∈ [mf , Mf ], yα,m+1 (τ, t, x, ρ) ∈ [mf , Mf ], for all τ ∈
[t, T ].
In this way, the functional cost becomes J(t, x, α(·)) = ym+1 (T ) + Ψ (y(T ))
and the function v, defined as
v(t, x, ρ) = inf yα,m+1 (T , t, x, ρ) + Ψ yα (T , t, x) : α(·) ∈ Ꮽ ,
(3.5)
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S. C. DI MARCO AND R. L. V. GONZÁLEZ
where t ∈ [0, T ], x ∈ Rm , and ρ ∈ [mf , Mf ], can be seen as the optimal cost of
an ordinary optimal control problem, that is, a problem with pure final cost.
3.2. Properties of the optimal cost v. The following properties bring some
relations between the optimal costs u and v of the original and auxiliary problems. They are almost evident and can be proved without difficulties using the
definitions of the original and auxiliary problems described above. Here, we
omit the complete proofs for the sake of brevity and we only sketch the lines
of the argument.
(1) Let mf (t, x) := mina∈A f (t, x, a), then, for all t ∈ [0, T ] and for all x ∈ Rm ,
u(t, x) = v t, x, mf (t, x) .
(3.6)
(2) We have v(t, x, Mf ) = Mf + û(t, x), where û is the optimal cost of the opti
x, α(·)) =
mal control problem where the functional to be minimized is J(t,
Ψ (y(T )).
In this case ym+1 (s) = Mf , for all s ∈ [t, T ]. Then, by replacing this equality
in (3.5), we have
v t, x, Mf = Mf + inf Ψ y(T ) = Mf + û(t, x).
α(·)∈Ꮽ
(3.7)
(3) If ρ1 > ρ2 , then
v t, x, ρ1 ≥ v t, x, ρ2 .
(3.8)
Let ρ1 > ρ2 , then
yα,m+1 T , t, x, ρ1 = max ρ1 , hα (t, T )
≥ max ρ2 , hα (t, T ) = yα,m+1 T , t, x, ρ2 .
(3.9)
By replacing this inequality in (3.5), it results that v(t, x, ρ1 ) ≥ v(t, x, ρ2 ).
(4) The optimal cost v is bounded in [0, T ] × Rm × [mf , Mf ].
This property follows from the definition of v and the boundedness of f
and Ψ .
(5) The optimal cost v is Lipschitz continuous with respect to the variables t,
x, and ρ.
This property follows from the hypotheses verified by f , g, and Ψ . The proof
can be obtained essentially with the techniques used in [21].
Note 3.4. As we have pointed out in Note 2.1, it is possible to consider f
and Ψ uniformly continuous with respect to their variables. It also results that
v is continuous with respect to the variables t, x, and ρ and similar results
can be obtained.
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
4523
3.3. Dynamical programming equation. In this new problem with the state
augmented by the variable ρ, the dynamical programming equation is given by
v(t, x, ρ) =
inf
α(·)∈L∞ ((t,s);A)
v s, y(s), ym+1 (s) ,
∀s ∈ (t, T ).
(3.10)
We also have the final condition
v(T , x, ρ) = max ρ, min f (T , x, a) + Ψ (x).
a∈A
(3.11)
Both relations are almost evident and can be proved without difficulties using the classical argument of dynamic programming and the definition of the
auxiliary problem described above. Here, we omit the complete proofs for the
sake of brevity.
4. Hamilton-Jacobi-Bellman equation
4.1. Preliminaries. We define
Hρ (t, x, ρ, ∇v) =
∂v
(t, x, ρ),
∂ρ
∂v
∂v
(t, x, ρ) +
(t, x, ρ)g(t, x, a) : f (t, x, a) < ρ ,
∂t
∂x
∂v
∂v
(t, x, ρ) +
(t, x, ρ)g(t, x, a) : f (t, x, a) ≤ ρ .
H∗ (t, x, ρ, ∇v) = inf
∂t
∂x
(4.1)
H(t, x, ρ, ∇v) = inf
Taking into account the dynamical programming equation, we obtain the following differential formulation. Strictly, it takes the following form:
min Hρ (t, x, ρ, ∇v), H(t, x, ρ, ∇v) ≥ 0,
(4.2)
min Hρ (t, x, ρ, ∇v), H∗ (t, x, ρ, ∇v) ≤ 0
with the final condition
v(T , x, ρ) = max ρ, min f (T , x, a) + Ψ (x),
a∈A
(4.3)
for all (x, ρ) ∈ Rm × [mf , Mf ], and the boundary condition
v t, x, Mf = Mf + û(t, x),
(4.4)
for all (t, x) ∈ [0, T ] × Rm , where û is the optimal cost of the optimal control
x, α(·)) = Ψ (y(T )).
problem where the functional to be minimized is J(t,
Note 4.1. It is clear that H ≥ H∗ . It can easily be proved that H∗ is the lower
semicontinuous envelope of H, that is,
H∗ (t, x, ρ, q) = lim inf H t , x , ρ , q : t − t < ε,
ε→0
(4.5)
x − x < ε, ρ − ρ < ε, q − q < ε .
4524
S. C. DI MARCO AND R. L. V. GONZÁLEZ
Note 4.2. When f ≡ 0, we have the Mayer problem. In this case, mf = Mf =
0 and the solution of the problem is given by (3.7) (with Mf = 0). So, ρ ≡ 0 and
the function û(t, x) is obtained by solving the classical HJB equation
inf
a∈A
∂v
∂v
(t, x, 0) +
(t, x, 0)g(t, x, a) = 0
∂t
∂x
(4.6)
with final condition v(T , x, 0) = Ψ (x), for all x ∈ Rm .
Note 4.3. When Ψ ≡ 0, we have the optimal control problem treated by Barron and Ishii [10]. We define ρ(t, x) = max{ρ ∈ [mf , Mf ] : v(t, x, ρ) = u(t, x)}.
Then, in case ρ > ρ(t, x), we have v(t, x, ρ) = ρ. In case ρ ≤ ρ(t, x), it follows
that v(t, x, ρ) = u(t, x). For the value ρ = u(t, x), it can be proved that the
system (4.2) becomes
∂u
∂u
(t, x) +
(t, x)g(t, x, a) : f (t, x, a) < u(t, x) ≥ 0,
a∈A
∂t
∂x
∂u
∂u
(t, x) +
(t, x)g(t, x, a) : f (t, x, a) ≤ u(t, x) ≤ 0,
inf
a∈A
∂t
∂x
inf
(4.7)
which is the HJB equation presented by Barron and Ishii [10]. Obviously, the
final condition is u(T , x) = v(T , x, mf (T , x)) = mf (T , x) = mina∈A f (T , x, a).
4.2. Viscosity solution. We say that a function v is a solution in the viscosity sense of system (4.2) when it is both a subsolution and a supersolution in
the viscosity sense of (4.2). These concepts are defined as follows.
In order to simplify the writing we define Ω = (0, T ) × Rm × (mf , Mf ). We
also give the following definitions.
(1) The function w is a subsolution in the viscosity sense if
(i) w is continuous on Ω;
(ii) w is upper-bounded on Ω;
(iii) w verifies the following final condition:
w(T , x, ρ) ≤ max ρ, min f (T , x, a) + Ψ (x),
a∈A
(4.8)
for all (x, ρ) ∈ Rm × [mf , Mf ];
(iv) w verifies the following upper condition: w(·, ·, Mf ) is a viscosity
solution of
H∗ t, x, Mf , ∇w = 0,
(4.9)
with final condition w(T , x, Mf ) ≤ Mf + Ψ (x), for all x ∈ Rm ;
(v) given (t, x, ρ) ∈ Ω and φ ∈ C 1 (Ω) such that φ−w has a minimum
in (t, x, ρ) in a neighborhood ᏺ(t, x, ρ), it results that
min Hρ (t, x, ρ, ∇φ), H(t, x, ρ, ∇φ) ≥ 0.
(4.10)
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
(2) The
(i)
(ii)
(iii)
function z is a supersolution in the viscosity sense if
z is continuous on Ω;
z is lower-bounded on Ω;
z verifies the following final condition:
z(T , x, ρ) ≥ max ρ, min f (T , x, a) + Ψ (x),
a∈A
4525
(4.11)
for all (x, ρ) ∈ Rm × [mf , Mf ];
(iv) z verifies the following upper condition: z(·, ·, Mf ) is a viscosity
solution of
H∗ t, x, Mf , ∇z = 0
(4.12)
with final condition z(T , x, Mf ) ≥ Mf + Ψ (x), for all x ∈ Rm ;
(v) given (t, x, ρ) ∈ Ω and φ ∈ C 1 (Ω) such that φ − z has a maximum
in (t, x, ρ) in a neighborhood ᏺ(t, x, ρ), it results that
min Hρ (t, x, ρ, ∇φ), H∗ (t, x, ρ, ∇φ) ≤ 0.
(4.13)
4.3. The optimal cost as a viscosity solution
Theorem 4.4. The optimal cost v is a solution in the viscosity sense of the
system (4.2).
Proof. The final condition is trivially verified from (3.11) and in Section 3.2
we have seen that v is continuous, bounded, and nondecreasing with respect
to ρ.
From (3.3), for ρ = Mf , the variable ρ remains constant and so, we are dealing
with an ordinary optimal control problem. Then, with classical arguments (see
[1, 20]), we obtain that v verifies the upper condition (4.9).
Now, it remains to prove the last condition of the subsolution’s definition
and the last one of the supersolution’s definition.
First, we will prove that v is a subsolution of (4.2).
Let (t, x, ρ) ∈ Ω and φ ∈ C 1 (Ω) such that φ − v has a minimum in (t, x, ρ)
in a neighborhood ᏺ(t, x, ρ).
Since v is nondecreasing in its third variable, we obtain
v(t, x, ρ) ≤ v(t, x, ρ + η).
(4.14)
By the minimality of φ − v at (t, x, ρ), it follows that
φ(t, x, ρ + η) − φ(t, x, ρ) ≥ v(t, x, ρ + η) − v(t, x, ρ) ≥ 0.
(4.15)
Dividing by η and taking limit when η goes to zero, we have
∂φ
(t, x, ρ) ≥ 0.
∂ρ
(4.16)
4526
S. C. DI MARCO AND R. L. V. GONZÁLEZ
Then,
Hρ (t, x, ρ, ∇φ) ≥ 0.
(4.17)
We prove that H(t, x, ρ, ∇φ) ≥ 0. Let a ∈ A such that f (t, x, a) < ρ and let
{αn } be a control sequence such that αn (s) = a for all s ∈ [t, t + n−1 ].
From (3.10), we have
−1
−1
.
v(t, x, ρ) ≤ v t + n , yn t + n
, max ρ, ess sup f s, yn (s), αn (s)
s∈[t,t+n−1 ]
(4.18)
Since
lim ess sup f s, yn (s), αn (s) = f (t, x, a),
n→∞
s∈[t,t+n−1 ]
∀n ≥ n0 ,
(4.19)
we get
max ρ, ess sup f s, yn (s), αn (s) = ρ,
(4.20)
s∈[t,t+n−1 ]
and then, from (4.18), we obtain
v(t, x, ρ) ≤ v t + n−1 , yn t + n−1 , ρ .
(4.21)
By the minimality of φ − v at (t, x, ρ), it follows that
φ t + n−1 , yn t + n−1 , ρ − φ(t, x, ρ)
≥ v t + n−1 , yn t + n−1 , ρ − v(t, x, ρ) ≥ 0.
(4.22)
Since
t+n−1
yn t + n−1 − x
=
lim
n
g s, y(s), αn (s) ds = g(t, x, a),
n→∞
n→∞
n−1
t
lim
(4.23)
dividing by n−1 in (4.22) and taking limit when n → ∞, we have
∂φ
∂φ
(t, x, ρ) +
(t, x, ρ)g(t, x, a) ≥ 0.
∂t
∂x
(4.24)
Then, since a is arbitrary, it results that
H(t, x, ρ, ∇φ) ≥ 0.
(4.25)
So, (4.17) and (4.25) imply that v is subsolution.
Now, we will show that v is a supersolution of (4.2).
Let (t, x, ρ) ∈ Ω and φ ∈ C 1 (Ω) such that φ − v has a maximum at (t, x, ρ)
in a neighborhood ᏺ(t, x, ρ).
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
4527
We will prove by reductio ad absurdum that
min Hρ (t, x, ρ, ∇φ), H∗ (t, x, ρ, ∇φ) ≤ 0;
(4.26)
so, we assume that there exists η > 0 such that
min Hρ (t, x, ρ, ∇φ), H∗ (t, x, ρ, ∇φ) ≥ η > 0.
(4.27)
Let {αn } be a minimizing control sequence such that
v(t, x, ρ)+
1
−1
−1
≥
v
t+n
,
y
f
s,
y
(s),
α
(s)
.
t+n
,
max
ρ,
ess
sup
n
n
n
n2
s∈[t,t+n−1 ]
(4.28)
We define
f
= lim ess sup f s, yn (s), αn (s) .
(4.29)
n→∞ s∈[t,t+n−1 ]
We suppose that f
> ρ. Taking limit in (4.28), it follows that v(t, x, ρ) ≥
v(t, x, f
). By the monotony of v, it results that v(t, x, ρ) = v(t, x, f
) and
v(t, x, ρ) = v(t, x, ) for all ∈ [ρ, f
]. This implies that
0 = v(t, x, ) − v(t, x, ρ) ≥ φ(t, x, ) − φ(t, x, ρ)
(4.30)
and then (∂φ/∂ρ)(t, x, ρ) ≤ 0. Inequality (4.27) implies that (∂φ/∂ρ)(t, x, ρ) =
Hρ (t, x, ρ, ∇φ) ≥ η. In consequence, η ≤ 0, which is absurd.
We suppose that f
≤ ρ. We will analyze the effect of this condition on the
following relation (valid by virtue of (4.27)):
H∗ (t, x, ρ, ∇φ) ≥ η > 0.
(4.31)
As f
≤ ρ, then, for all ε > 0, there exists nε such that, for all n ≥ nε ,
ess sup f s, yn (s), αn (s) < ρ + ε.
(4.32)
s∈[t,t+n−1 ]
Eventually, by redefining the controls αn (·) in zero-measure sets, it is possible
to affirm that, for all n ≥ nε , for all s ∈ [t, t + n−1 ],
f s, yn (s), αn (s) < ρ + ε.
(4.33)
Since f is Lipschitz-continuous, there exists nε such that, for all n ≥ nε and
for all s ∈ [t, t+n−1 ],
f t, x, αn (s) < ρ + ε + Lf 1 + Mg n−1 < ρ + 2ε.
(4.34)
4528
S. C. DI MARCO AND R. L. V. GONZÁLEZ
We consider the set of controls Zε given by
Zε = a ∈ A : f (t, x, a) ≤ ρ + 2ε .
(4.35)
Then, for any n ≥ nε and s ∈ [t, t + n−1 ], it results that αn (s) ∈ Zε .
As (t, x, ρ) is a maximum point for the function φ − v, we have
φ t + n−1 , yn t + n−1 , ρ − φ(t, x, ρ)
≤ v t + n−1 , yn t + n−1 , ρ − v(t, x, ρ).
(4.36)
Since v is nondecreasing in “ρ,” from (4.28), we get
v t + n−1 , yn t + n−1 , ρ
−1
−1
≤ v t + n , yn t + n
, max ρ, ess sup f s, yn (s), αn (s)
s∈[t,t+n−1 ]
(4.37)
≤ v(t, x, ρ) + n−2 ,
thus,
φ t + n−1 , yn t + n−1 , ρ − φ(t, x, ρ) ≤ n−2 .
(4.38)
It is possible to prove that there exists g ∈ Co(g(t, x, Zε )), where Co(E) is the
convex hull of the set E such that (eventually using a suitable subsequence)
t+n−1
lim n
n→∞
t
g s, yn (s), αn (s) ds = g,
(4.39)
and so, by virtue of (4.38), we get
φ t + n−1 , yn t + n−1 , ρ − φ(t, x, ρ)
lim
n→∞
n−1
∂φ
∂φ
=
(t, x, ρ) +
(t, x, ρ)g ≤ 0.
∂t
∂x
(4.40)
On the other hand, as Zε is closed, we have
Co g t, x, Zε = Co g t, x, Zε .
(4.41)
As g ∈ Co(g(t, x, Zε )) for all ε > 0, we get that g ∈ Co(g(t, x, Z0 )). Then, there
exists a probability measure µ(·) with support in Z0 such that
g=
g(t, x, a)dµ(a).
(4.42)
Z0
From (4.31), if a ∈ Z0 , then
∂φ
∂φ
(t, x, ρ) +
(t, x, ρ)g(t, x, a) ≥ η > 0.
∂t
∂x
(4.43)
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
4529
Using this inequality and the integral expression (4.42) in (4.40), we have
∂φ
∂φ
(t, x, ρ) +
(t, x, ρ)g
∂t
∂x
∂φ
∂φ
(t, x, ρ) +
(t, x, ρ)g(t, x, a) dµ(a) ≥ η.
=
∂t
∂x
Z0
0≥
(4.44)
This inequality contradicts the initial assumption (4.27) and so, we obtain by
reductio ad absurdum that v is a supersolution.
4.4. Uniqueness of the viscosity solution
Theorem 4.5. There is a unique viscosity solution of system (4.2).
Proof. Let w be a subsolution and z a supersolution of (4.2).
We will prove by reductio ad absurdum that w ≤ z.
In the case mf = Mf , the result is obvious because in that case the conditions
(4.8) and (4.9) verified by w and the conditions (4.9) and (4.11) verified by z
imply that w ≤ z (the proof now follows classical arguments (see [1, 20]) and
it is here omitted for the sake of brevity). So, we will consider only the case
mf < Mf .
We suppose that
r := sup w(t, x, ρ) − z(t, x, ρ) : (t, x, ρ) ∈ Ω > 0,
(4.45)
then, there exists (tr , xr , ρr ) ∈ Ω such that
r
w tr , xr , ρr − z tr , xr , ρr > .
2
(4.46)
M
, in [0, T ] × Rm × mf , Mf ,
2
M
z(t, x, ρ) ≥ − , in [0, T ] × Rm × mf , Mf .
2
(4.47)
Let M > 0 such that
w(t, x, ρ) ≤
For each > 0 and γ > 0, we consider the hump function
√ 2
|t − s|2 x − y2 ρ − ζ + −
−
Φ(s, y, ζ, t, x, ρ) = −
2
2
2
Mf − mf
T
−1 −σ
−σ
−1
t
ρ − mf
t
r
1−
− 2Mξ γ x − xr .
−
8
T
(4.48)
4530
S. C. DI MARCO AND R. L. V. GONZÁLEZ
The elements of Φ have the following properties:
(1) ξ(·) ∈ C 1 (R) verifies

0,
ξ(τ) =
1,
∀τ ∈ [0, 1];
∀τ ≥ 3;
(4.49)
for |ξ (τ)| ≤ 1, for all τ ∈ R;
(2) σ < min{r tr /(8(T − tr )), r (ρr − mf )/(8(Mf − mf ))}.
We define the function
φ(s, y, ζ, t, x, ρ) = w(s, y, ζ) − z(t, x, ρ) + Φ(s, y, ζ, t, x, ρ).
(4.50)
As w is a subsolution, it must verify Hρ (t, x, ρ, ∇w) ≥ 0. This condition implies that, for all ρ ∈ [mf , Mf ) and for all > 0 such that ρ + ≤ Mf , we
have w(s, y, ρ + ) ≥ w(s, y, ρ) (the proof of this intuitive property is essentially contained in [23] and it is here omitted). Since ρr < Mf , we can suppose,
√
without loss of generality, that is small enough to verify ρr + < Mf , and
so
√ w tr , xr , ρr + − w tr , xr , ρr ≥ 0.
(4.51)
Now, taking into account that
r
tr
r
1−
≤ ,
8
T
8
T − tr
σ
tr
r
≤ ,
8
σ
Mf − ρr
ρr − mf
≤
r
,
8
(4.52)
from (4.46), (4.48), (4.50), (4.51), and (4.52), we have
√
φ tr , xr , ρr + , tr , xr , ρr
3r r
√ r
+ ≥ > 0.
≥ w tr , xr , ρr + − w tr , xr , ρr −
8
2
8
(4.53)
We consider a maximizing sequence (sµ , yµ , ζµ , tµ , xµ , ρµ ) such that
lim φ sµ , yµ , ζµ , tµ , xµ , ρµ = sup φ(s, y, ζ, t, x, ρ).
µ→∞
(4.54)
Ω×Ω
Let (sµ , yµ , ζµ , tµ , xµ , ρµ ) be an element of this maximizing sequence. Therefore, for all µ ≥ µ0 suitably chosen, we have
r
√
φ sµ , yµ , ζµ , tµ , xµ , ρµ ≥ φ tr , xr , ρr + , tr , xr , ρr ≥ > 0.
8
(4.55)
Since (4.47) implies that
√ w sµ , yµ , ρµ + − z tµ , xµ , ρµ ≤ M,
(4.56)
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
4531
from (4.48), (4.55), and (4.56), we get
xµ − yµ 2 ρµ − ζµ − √2
T − tµ
+
+
+σ
2
2
tµ
Mf − ρµ
+ 2Mξ γ xµ − xr +σ
ρµ − mf
r
≤ M − < M.
8
t µ − s µ 2
2
(4.57)
Then, inequality (4.57) and the properties of ξ imply that
σ Mf − mf
σT
+ mf < ρµ ,
< tµ ,
M +σ
M +σ
√
√
tµ − sµ < M,
xµ − yµ < M,
√
ρµ − ζµ + √ < M,
(4.58)
xµ − xr ≤ 3 .
γ
Moreover, in the case mf < Mf , the conditions (4.8) and (4.9) verified by w and
the conditions (4.9) and (4.11) verified by z imply that
w s, y, Mf ≤ z s, y, Mf ,
∀(s, y) ∈ (0, T ) × Rm
(4.59)
(the proof is easy (see [1, 20]) and it is here omitted).
We denote
4
Br = x ∈ Rm : x − xr ≤
.
γ
(4.60)
From (4.59) and the continuity of w and z, we have that there exists δr > 0
such that if xµ ∈ Br , yµ ∈ Br , and
tµ − sµ ≤ δr ,
ζµ ≥ Mf − δr ,
xµ − yµ ≤ δr ,
(4.61)
ρµ ≥ Mf − δr ,
(4.62)
then
r
.
φ sµ , yµ , ζµ , tµ , xµ , ρµ ≤ w sµ , yµ , ζµ − z tµ , xµ , ρµ ≤
16
(4.63)
From (4.58), we have that, for small enough, conditions (4.61) are verified,
xµ ∈ Br , and yµ ∈ Br . So, if (4.62) is verified, we get a contradiction between
(4.55) and (4.63). In consequence, we get
ζµ < Mf − δr
or
ρµ < Mf − δr .
(4.64)
4532
S. C. DI MARCO AND R. L. V. GONZÁLEZ
Finally, this condition and (4.58) imply—again for small enough—that
ζ µ < Mf −
δr
,
2
ρµ < Mf −
δr
.
2
(4.65)
Consequently, (sµ , yµ , ζµ , tµ , xµ , ρµ ) is a bounded sequence which verifies (4.58).
Then, we can suppose, without loss of generality, that
sµ , yµ , ζµ , tµ , xµ , ρµ
→ s , y , ζ , t , x , ρ ∈ Ω × Ω.
(4.66)
In addition, from (4.58) and (4.65), we have
σ Mf − mf
σT
+ mf ≤ ρ ,
≤ t ,
M
M
√
√
x − y ≤ M,
t − s ≤ M,
δr
,
2
x − xr ≤ 3 ,
γ
ζ ≤ Mf −
δr
,
2
√
ρ − ζ + √ ≤ M,
ρ ≤ Mf −
(4.67)
and so,
δr
δr
× (0, T )×Br × mf , Mf −
.
s , y , ζ , t , x , ρ ∈ (0, T )×Br × mf , Mf −
2
2
(4.68)
From (4.54), we get
φ s , y , ζ , t , x , ρ = sup φ(s, y, ζ, t, x, ρ)
(4.69)
Ω×Ω
because w and z are continuous in Ω, and then w(s, y, ζ) − z(t, x, ρ) is continuous in Ω × Ω.
We define
ψ : (0, T ] × Br × mf , Mf × (0, T ] × Br × mf , Mf → R,
Mf − mf
−1
ψ(s, y, ζ, t, x, ρ) = w(s, y, ζ) − z(t, x, ρ) − σ
(4.70)
ρ − mf
r
t
T
−1 −
1−
− 2Mξ γ x − xr ,
−σ
t
8
T
√
δr
,
Sψ = sup ψ t, x, ρ + , t, x, ρ : (t, x, ρ) ∈ (0, T ] × Br × mf , Mf −
2
(4.71)
4533
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
R+
0
χw,z :
→
χw,z (η) = sup ψ(s, y, ζ, t, x, ρ)
R+
0,
δr
,
− Sψ : (t, x, ρ) ∈ [0, T ] × Br × mf , Mf −
2
δr
,
(s, y, ζ) ∈ [0, T ] × Br × mf , Mf −
2
√ |s − t| + y − x + ζ − ρ − ≤ η .
(4.72)
It is obvious that χw,z is an increasing function verifying χw,z (0) = 0. In addition, from the continuity of ψ in (0, T ]×Br ×(mf , Mf ]×(0, T ]×Br ×(mf , Mf ],
we get limη→0 χw,z (η) = 0.
From the optimality of (s , y , ζ , t , x , ρ ), we have for all (t, x, ρ) ∈ (0, T ]×
√
Br × (mf , Mf − δr /2] such that ρ + ∈ (mf , Mf ],
√
φ s , y , ζ , t , x , ρ ≥ φ t, x, ρ + , t, x, ρ
√
= ψ t, x, ρ + , t, x, ρ .
(4.73)
From this inequality and taking into account (4.50) and (4.70), we get
t − s 2
2
+
x − y 2
+
ρ − ζ − √2
2
2
√
≤ ψ s , y , ζ , t , x , ρ − ψ t, x, ρ + , t, x, ρ .
(4.74)
From this inequality and (4.71), we get
t − s 2
2
+
x − y 2
2
+
ρ − ζ − √2
2
≤ ψ s , y , ζ , t , x , ρ − Sψ ,
(4.75)
and then, from (4.67) and (4.72),
t − s 2
2
+
x − y 2
2
+
ρ − ζ − √2
2
√
≤ χw,z 3 M .
(4.76)
From (4.76), we get
√
t − s ≤ χw,z 3 M ,
√
x − y ≤ χw,z 3 M ,
√
ρ − ζ + √ ≤ χw,z 3 M .
(4.77)
We define the function φ1 ∈ C 1 ((0, T ) × Rm × (mf , Mf )) as follows:
φ1 (t, x, ρ) = w s , y , ζ + Φ s , y , ζ , t, x, ρ .
(4.78)
4534
S. C. DI MARCO AND R. L. V. GONZÁLEZ
From (4.50) and (4.69), φ1 − z has a maximum in (t , x , ρ ), then, as z is a
supersolution, it follows that
min Hρ t , x , ρ , ∇φ1 , H∗ t , x , ρ , ∇φ1 ≤ 0.
(4.79)
If
min Hρ t , x , ρ , ∇φ1 , H∗ t , x , ρ , ∇φ1 = Hρ t , x , ρ , ∇φ1 ,
(4.80)
then
Hρ t , x , ρ , ∇φ1 ≤ 0.
(4.81)
∂φ1
∂Φ (t, x, ρ) =
s , y , ζ , t, x, ρ
∂ρ
∂ρ
√ Mf − mf
ρ − ζ + +σ = −2
2 ,
2
ρ − mf
(4.82)
∂φ1 Hρ t , x , ρ , ∇φ1 =
t , x , ρ
∂ρ
√ M − mf
ρ − ζ + f
+
σ
.
= −2
2
ρ − mf )2
(4.83)
Since
we have
On the other hand, by defining φ2 ∈ C 1 ((0, T ) × Rm × (mf , Mf )),
φ2 (s, y, ζ) = z t , x , ρ − Φ s, y, ζ, t , x , ρ ,
(4.84)
from (4.50) and (4.69), we have φ2 − w has a minimum in (s , y , ζ ). As w is
a subsolution, we get
Hρ s , y , ζ , ∇φ2 ≥ 0.
(4.85)
Since
√ ρ − ζ + ∂Φ ∂φ2
(s, y, ζ) = −
s, y, ζ, t , x , ρ = −2
,
∂ζ
∂ζ
2
(4.86)
we have
√ ∂φ2 ρ − ζ + s , y , ζ = −2
.
(4.87)
Hρ s , y , ζ , ∇φ2 =
∂ζ
2
√
Thus, from (4.85), we get 2(ρ −ζ + )/2 ≤ 0. Now, replacing this inequality
in (4.83), we obtain
√ Mf − mf
ρ − ζ + +σ (4.88)
Hρ t , x , ρ , ∇φ1 = −2
2 > 0.
2
ρ − mf
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
4535
This inequality contradicts (4.81), then it is only possible that the following
equality holds:
min Hρ t , x , ρ , ∇φ1 , H∗ t , x , ρ , ∇φ1 = H∗ t , x , ρ , ∇φ1 .
(4.89)
In consequence, it must follow that
H∗ t , x , ρ , ∇φ1 ≤ 0.
(4.90)
Let a ∈ A such that f (t , x , a) ≤ ρ and
∂φ1 ∂φ1 t , x , ρ +
t , x , ρ g t , x , a ≤ 0.
∂t
∂x
(4.91)
From (4.77), we get
√
f s , y , a ≤ f t , x , a + 2Lf χw,z 3 M
√
≤ ρ + 2Lf χw,z 3 M .
(4.92)
√
√
Also, from (4.77), ρ ≤ ζ + χw,z (3 M) − and then (for small enough),
√
√
√
f s , y , a ≤ ζ + χw,z 3 M − + 2Lf χw,z 3 M
√
≤ ζ −
< ζ .
2
(4.93)
As w is a subsolution, we have
H s , y , ζ , ∇φ2 ≥ 0,
(4.94)
and so, by definition of H, we have
∂φ2 ∂φ2 s , y , ζ +
s , y , ζ g s , y , a ≥ 0.
∂s
∂y
(4.95)
The derivatives of Φ are
∂Φ
(t − s)
T
r
(s, y, ζ, t, x, ρ) = −2
,
+σ 2 +
∂t
2
t
8T
∂Φ
(x − y)
x − xr x − xr ,
(s, y, ζ, t, x, ρ) = −2
−
2γMξ
γ
x − xr ∂x
2
∂Φ
(t − s)
,
(s, y, ζ, t, x, ρ) = 2
∂s
2
∂Φ
(x − y)
,
(s, y, ζ, t, x, ρ) = 2
∂y
2
(4.96)
4536
S. C. DI MARCO AND R. L. V. GONZÁLEZ
and then,
∂φ2
t − s
∂Φ ,
(s, y, ζ) = −
t , x , ρ , s, y, ζ = −2
∂s
∂s
2
∂Φ x − y
∂φ2
(s, y, ζ) = −
,
t , x , ρ , s, y, ζ = −2
∂y
∂y
2
∂φ1
T
r
∂Φ t − s
+σ 2 +
(t, x, ρ) =
t, x, ρ, s , y , ζ = −2
,
∂t
∂t
2
t
8T
∂Φ
∂φ1
(t, x, ρ) =
(t, x, ρ, s , y , ζ
∂x
∂x
x − y
x − xr x − xr .
γ
= −2
−
2γMξ
x − xr 2
(4.97)
Using these expressions in (4.91) and (4.95), we have
0<
r
T
x − y , g t , x , a − g s , y , a
≤ −σ 2 + 2
8T
2
t
x − x r , g t , x , a
+ 2γMξ γ x − xr x − xr 2Lg ≤ 2 x − y x − y + t − s + 2γM g t , x , a √
≤ 4Lg χw,z 3 M + 2γMMg .
(4.98)
Letting successively and γ go to zero, we have r ≤ 0 which contradicts inequality (4.45). Then, z(t, x, ρ) ≥ w(t, x, ρ) for all (t, x, ρ) ∈ Ω. Finally, taking
into account that any solution is at the same time a subsolution and a supersolution, we obtain the uniqueness of solution.
5. Conclusion. In this paper, we have analyzed some issues concerning the
viscosity solution of an HJB system associated to a minimax optimal control
problem with finite horizon and additive final cost.
In the first step, we have introduced an auxiliary problem which generalizes
the original one. It is possible to get the solution of the original problem in
terms of the solution of the auxiliary problem.
For this auxiliary problem, a dynamical programming principle (DPP) holds.
In relation with this DPP, we have presented an HJB system defined in terms
of a discontinuous Hamiltonian and we have proved that the optimal cost of
the auxiliary problem is the unique viscosity solution of this HJB system.
Although this problem could be seen as a particular (deterministic) case of
those treated by Barles, Daher, and Romano [4], there are several differences
between the results contained in [4] and those presented in this paper, among
them are the following ones.
(1) We present direct proofs of the existence and uniqueness of solution
without requiring the treatment of a sequence of Lp problems. They are obtained through a different methodology based in control theory.
ON A SYSTEM OF HAMILTON-JACOBI-BELLMAN INEQUALITIES . . .
4537
(2) In our paper the HJB equation is deduced without the hypothesis Im(f )
= [mf , Mf ]. So, our HJB equation is valid, in particular, for the cases where the
controls take a finite number of values. In those special cases, the Lp penalization technique used by Barles, Daher, and Romano does not allow deducing
it.
(3) The HJB system here obtained is simpler than that one presented in
[4] because no conditions are required at the lower boundary (0, T ) × Rm ×
{mf }.
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Silvia C. Di Marco: CONICET, Instituto de Matematica Beppo Levi, FCEIA, Universidad
Nacional de Rosario, Pellegrini 250, Rosario 2000, Argentina
E-mail address: dimarco@fceia.unr.edu.ar
Roberto L. V. González: CONICET, Instituto de Matematica Beppo Levi, FCEIA, Universidad Nacional de Rosario, Pellegrini 250, Rosario 2000, Argentina
E-mail address: rlvgonza@fceia.unr.edu.ar